A classification of triangular Riemann surfaces with $2p^2$ automorphisms

Technical Summary: A Classification of Triangular Riemann Surfaces with $2p^2$ Automorphisms

Problem Statement
This article addresses the classification of compact Riemann surfaces of genus $g \geq 2$ that admit a "triangular" action by a finite group of order $2p^2$, where $p$ is an odd prime. A triangular action is defined as an action where the quotient space has genus 0 and the covering map ramifies over exactly three points (signature $(0; m_1, m_2, m_3)$). Such surfaces are termed quasiplatonic. The authors aim to determine the necessary and sufficient conditions for abstract groups of order $2p^2$ to act triangularly, classify the resulting surfaces up to isomorphism, compute their genera and full automorphism groups, and provide explicit algebraic models. This work extends previous classifications, such as those for groups of order $pq$ (Streit and Wolfart) and cyclic/abelian groups, to the more intricate case of order $2p^2$, where multiple non-isomorphic groups and multiple distinct actions per group arise.

Methodology
The authors employ the combinatorial theory of Fuchsian groups and the study of inclusions among them, a framework developed by Singerman. The core methodology involves:

1. Signature Analysis: Identifying which of the five non-isomorphic groups of order $2p^2$ ($D_{p^2}$, $Z_{p^2} \rtimes Z_2$, $Z_{2p} \times Z_p$, $D_p \times Z_p$, and $Z_{2p^2}$) can admit a triangular action. This is achieved by analyzing the generators required to surjectively map a Fuchsian group of signature $(0; m_1, m_2, m_3)$ onto the target group.
2. Surface-Kernel Epimorphisms (SKEs): Constructing epimorphisms $\Phi: \Gamma \to G$ from Fuchsian groups $\Gamma$ to the target groups $G$. The kernel of such a map defines the Riemann surface $S \cong \mathbb{H}/\ker(\Phi)$.
3. Extension of Actions: Utilizing Singerman's list of maximal signatures and the criteria established by Bujalance, Cirre, and Conder to determine if a given action extends to a larger automorphism group. This involves checking for inclusions of Fuchsian groups and the existence of specific involutions in the automorphism group that satisfy extension conditions.
4. Algebraic Modeling: Describing the classified surfaces as cyclic covers of the projective line (cyclic $n$-gonal curves). The authors derive explicit affine plane curves of the form $y^n = \prod (x - \alpha_i)^{n_i}$ by analyzing branch values and rotation numbers of the automorphisms.
5. Isomorphism Classification: Determining the number of non-isomorphic surfaces by analyzing the orbits of SKEs under the action of the automorphism group of the Fuchsian group and the target group.

Key Contributions and Results
The paper provides a complete classification of compact Riemann surfaces admitting a triangular action of a group of order $2p^2$. The main findings are:

• Existence and Count:

  • The groups $D_{p^2}$ and $Z_p^2 \rtimes Z_2$ do not admit triangular actions for $g \geq 2$.
  • For $p \geq 5$, there are exactly $\frac{p^2 + 2p + 3}{2}$ pairwise non-isomorphic compact Riemann surfaces admitting such an action.
  • For $p = 3$, there are exactly 8 such surfaces.
  • The surfaces are distributed among the groups $Z_{2p} \times Z_p$, $D_p \times Z_p$, and $Z_{2p^2}$.

• Specific Classifications:

  • $Z_{2p} \times Z_p$: Admits a unique triangular action with signature $(2p, 2p, p)$. The resulting surface has genus $g = (p-1)^2$ and is isomorphic to the normalization of $y^p = x^{2p} - 1$. Its full automorphism group is $D_p \times Z_{2p}$.
  • $D_p \times Z_p$: Admits actions with signatures $(2, p, 2p)$ and $(2p, 2p, p)$.
    • Signature $(2, p, 2p)$ yields a unique surface of genus $\frac{(p-1)(p-2)}{2}$, isomorphic to $y^p = (x^p - 1)^2$, with full automorphism group $Z_p^2 \rtimes D_3$.
    • Signature $(2p, 2p, p)$ yields $\frac{p+1}{2}$ non-isomorphic surfaces of genus $(p-1)^2$. These are described by curves $y^p = (x^p - 1)^2(x^p + 1)^{\epsilon}$. Their full automorphism groups vary: $D_p^2 \rtimes Z_2$ for one case, $D_p \times Z_{2p}$ for another, and $D_p \times Z_p$ for the rest.
  • $Z_{2p^2}$: Admits actions with four distinct signatures: $(2, p^2, 2p^2)$, $(2p^2, 2p^2, p)$, $(2p, p^2, 2p^2)$, and $(2p^2, 2p^2, p^2)$.
    • These yield $1 + \frac{p-1}{2} + (p-1) + \frac{p^2 - 2p + 1}{2}$ distinct isomorphism classes.
    • Explicit algebraic models are provided for each case (e.g., $y^{p^2} = x^2 + 1$, $y^p = x^2(x^{2p}-1)^m$, etc.).
    • The full automorphism group is generally $Z_{2p^2}$, except for specific cases where it extends to $Z_{p^2} \rtimes D_4$.

• Field of Definition:

  • A significant result is that all identified Riemann surfaces are defined over the rational numbers ($\mathbb{Q}$). This is derived from the fact that their algebraic models are given by polynomials with rational coefficients.

• Hypermap Classification:

  • As a by-product, the authors classify orientably-regular hypermaps (or dessins d'enfants) with orientation-preserving automorphism groups of order $2p^2$. For $p \geq 5$, there are exactly $p^2 + p$ such non-equivalent hypermaps.

Significance
The paper claims significance in several areas:

1. Completeness: It provides a definitive classification for the specific order $2p^2$, a case more complex than the previously studied $pq$ order due to the existence of non-abelian groups and multiple action types.
2. Rationality: The result that all such surfaces are defined over $\mathbb{Q}$ contrasts with the general behavior of Riemann surfaces with automorphisms, which often require number fields. This reinforces the special arithmetic nature of quasiplatonic surfaces.
3. Algebraic Models: The explicit construction of affine plane curves for every isomorphism class allows for concrete computational study and connects the geometric classification to algebraic geometry.
4. Moduli Space Geometry: By identifying the singular locus of the moduli space $M_g$ corresponding to these specific group actions, the work contributes to understanding the geometry of the moduli space of Riemann surfaces.
5. Combinatorial Connection: The classification bridges the gap between the theory of Riemann surfaces and the combinatorics of hypermaps, offering a complete list of orientably-regular hypermaps for this specific automorphism order.

The authors emphasize that their results are derived through rigorous application of Fuchsian group theory and do not rely on unproven conjectures, providing a solid foundation for further study of the arithmetic and geometric properties of these surfaces.